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Question:

(a) Use Gauss' law to derive the expression for the electric field (→E) due to a straight uniformly charged infinite line of charge density λ (C/m). (b) Draw a graph to show the variation of E with perpendicular distance r from the line of charge. (c) Find the work done in bringing a charge q from perpendicular distance r1 to r2 (r2 > r1).

Solution:

A.Gauss' law states that total electric flux over the closed surface S is 1/ε₀ times the total charge (q) contained inside S.
The graph is as shown in figure
C. Work done in bringing charge : Vb−Va=∫r1r2 −Er dr =∫r1r2 λ/2πε₀r dr = λ/2πε₀ ln(r2/r1) so Vb−Va=W/work done=(Vb−Va)